Julia set

Let U be an open subset of the complex plane and let f:UU be analytic. Denote the n-th iterate of f by fn, i.e. f1=f and fn+1=ffn. Then the Julia set of f is the subset J of U characterized by the following property: if zJ then the restriction of {fnn} to any neighborhoodMathworldPlanetmath of z is not a normal family.

It can also be shown that the Julia set of f is the closure of the set of repelling periodic points of f. (Repelling periodic point means that, for some n, we have fn(z)=z and |f(z)|>1.)

A simple example is afforded by the map f(z)=z2; in this case, the Julia set is the unit circle. In general, however, things are much more complicated and the Julia set is a fractal.

From the definition, it follows that the Julia set is closed under f and its inversef(J)=J and f-1(J)=J. Topologically, Julia sets are perfect and have empty interior.

Title Julia set
Canonical name JuliaSet
Date of creation 2013-03-22 17:15:26
Last modified on 2013-03-22 17:15:26
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 7
Author rspuzio (6075)
Entry type Definition
Classification msc 28A80